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Jyri Pakarinen∗ and David T. Yeh† A Review of Digital ∗Department of Processing and Acoustics Techniques for Modeling Helsinki University of Technology P.O. Box 3000 FI-02015 TKK Finland Vacuum-Tube [email protected] †Center for Research Amplifiers in Music and Acoustics Department of Music Stanford University Stanford, California 94305-8180 USA [email protected]

Although semiconductor technologies have dis- as plug-ins so that the musician can placed vacuum-tube devices in nearly all fields of connect the guitar directly to the computer’s , vacuum tubes are still widely used in card, record the input tracks, add effects and/or professional guitar . A major reason for virtual instruments, and then compile the song as this is that electric-guitar amplifiers are typically a CD or upload it to the Internet. This is especially overdriven, that is, operated in such a way that the useful for home studios and small ad hoc recording output saturates. Vacuum tubes distort the signal in sessions, because it eliminates several tedious a different manner compared to solid-state electron- tasks of acoustic recording, such as setting up the ics, and human listeners tend to prefer this. This and recording equipment, selecting a might be because the distinctive tone of tube am- position, finding a recording room, etc. plifiers was popularized in the 1950s and 1960s by This article attempts to summarize real-time early bands, so musicians and listeners digital techniques for modeling guitar tube ampli- have become accustomed to tube . Some fiers. Although a brief overview was presented in studies on the perceptual aspects of vacuum-tube Pakarinen (2008), to the authors’ knowledge, there and solid-state distortion have been published (e.g., are no previous works that attempt a comprehensive Hamm 1973; Bussey and Haigler 1981; Santo 1994). survey of the topic. Because this topic is relatively Despite their acclaimed tone, vacuum-tube new and commercially active, most of the reference amplifiers have certain shortcomings: large size and material can be found in patents rather than aca- weight, poor durability, high consumption, demic publications. Judging from the large number high price, and often poor availability of spare parts. of amateur musicians and home-studio owners, as Thus, it is not surprising that many attempts have well as the huge number of discussion threads on been made to emulate guitar tube amplifiers using Internet forums, this topic is potentially interesting smaller and cheaper solid-state analog circuits (e.g., for a wide spectrum of readers. Thus, a conscious Todokoro 1976; Sondermeyer 1984). The next step choice has been made to try to survey the modeling in the evolution of tube-amplifier emulation has techniques at an abstracted level, without delving been to simulate the amplifiers using into the underlying mathematics or electric circuit and digital signal processors (DSP). analysis. A primary advantage of digital emulation is that This review is organized into four sections. We the same hardware can be used for modeling many first describe the sources of the nonlinearities in different tube amplifiers and effects. When a new circuits. Then, we review published model is to be added, new parameter values or methods for modeling the linear stages of guitar program code are simply uploaded to the device. amplifiers. The heart of this survey is the review Furthermore, amplifier models can be implemented of methods for nonlinear modeling. Finally we

Computer Music Journal, 33:2, pp. 85–100, Summer 2009 c 2009 Massachusetts Institute of Technology.

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2009.33.2.85 by guest on 26 September 2021 Figure 1. Physical construction (a) and electrical representation (b) of a tube. (Figure (a) is adapted from en.wikipedia.org/wiki/Vacuum tube.)

mention various other guitar-amplifier related technologies and present conclusions.

Vacuum-Tube Amplifiers

The purpose of this section is to present an overview of the operation of vacuum-tube amplifiers and to illustrate the complex nature of their important nonlinearities. An overview of vacuum tubes used in audio applications can be found in Barbour (1998), and a detailed tutorial on classic vacuum- tube circuits is provided in Langford-Smith (1954). The physical principles governing the operation of vacuum tubes are reviewed in Spangenberger (1948). Excellent Internet articles discussing the design of guitar tube amplifiers can be found online (e.g., at www.aikenamps.com and www.ax84.com). A typical guitar tube amplifier consists of a commonly used for signal rectification. Three- , a tone-control circuit (i.e., tone stack), terminal devices are known as and are a power amplifier, and a that couples primarily used in preamplifier circuits. Four- and to the load. The preamplifier magnifies five-terminal devices ( and , re- the relatively weak signal from the magnetic spectively) are used mainly for power amplification guitar pickups and provides buffering so that the purposes to drive a loudspeaker, for example. pickup response is not altered by the amplifier The operation of vacuum tubes is analogous to circuitry. The preamplifier is usually realized with water flow on a slope. First, the electrode termed triode tubes. The tone stack provides a typical V- the cathode is heated, and the process known as shaped for compensating the pickup’s thermionic emission acts like a pump that forms resonance at mid-, and it gives the a pool of electrons at the top of a hill. A second user additional tonal control. The power amplifier terminal called the plate (or anode) is at the bottom boosts the signal so that it is powerful enough of a slope. Electrons will flow from the cathode to to drive a loudspeaker. In the so-called all-tube the plate depending upon the relative height of the guitar amplifiers, both the pre- and power-amplifier plate, which is controlled by the applied circuits use tubes instead of in amplifying to it. Note that because a pump is at the cathode, the signal. Typically, these amplification circuits electrodes can never flow backward from the plate contain one or more tube stages, namely, circuit to the cathode even though the plate may be raised blocks that consist of a tube connected to resistive uphill of the cathode. This describes the rectification and capacitive (RC) components. behavior of a tube. The triode, illustrated in Figure 1, introduces a third terminal called the grid between the two Vacuum Tubes terminals. With the plate downhill of the cathode, the grid is like a raised barrier in the slope that Vacuum tubes, or thermionic valves, were invented limits the flow of electrons from the cathode to in the early for amplifying low-level volt- the plate. If this barrier controlled by the grid is age . Structurally, they consist of two or high enough, it stops the electron flow completely. more electrodes in a vacuum enclosed in a glass This water-flow analogy motivates the British term or metal shell. A two-terminal device is a diode, referring to vacuum tubes as “valves.”

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2009.33.2.85 by guest on 26 September 2021 Nonlinear Amplification of the input signal, the grid current Igk charges the and dynamically varies the bias point of The plate-to-cathode current is a nonlinear function the tube, leading to dynamically varying transient of both the grid-to-cathode and plate-to-cathode distortion characteristics. : Ipk = f (Vgk,Vpk). Note that a change in The cathode bypass capacitor retains memory of voltage on the grid causes a change in current the tube bias and responds slowly to rapid changes in flow between the cathode and plate. Amplification signal , causing signal history–dependent occurs when the change in current is converted to changes in distortion characteristics. Furthermore, a change in voltage by a large-valued load . there exist parasitic capacitances in the tube itself Although amplification is nominally linear around owing to the close proximity of its electrodes. The a central operating voltage known as the bias,at dominant effect, Miller capacitance, is a low-pass extreme signal levels, the amplified output will filter resulting from the amplified capacitance saturate. When the grid-to-cathode voltage Vgk is between plate and grid; this is discussed more very small, current flow cuts off sharply. Very large thoroughly in Aiken (1999a). Vgk causes the plate voltage to approach that of the cathode again, limiting the current and resulting in a nonlinearly saturating characteristic. To find the Amplifier Power Stage full nonlinear transfer characteristic from input to output requires the solution of a The power amplifier can use either a single-ended of implicit equations, because in a typical amplifier or push–pull topology. In the single-ended topology, circuit, Vpk depends on Ipk and vice versa. the signal is amplified in a single . This In guitar-amplifier circuits, the operating point tube conducts plate-to-cathode current during the (bias), defined in terms of current through the tube whole signal cycle (Class A ). Parallel tube device, is often set by a resistor connecting the stages can also be added if more output power is cathode terminal to ground. The resistor introduces required. into the circuit, and its value influences The push–pull topology, perhaps more commonly the shape of the input-output curve and determines used, consists of two identical sets of output tubes the offset about which the signal varies. Amplifier driven in opposite phases. The output of one set designs often include an AC bypass capacitor to is inverted and combined with the other through recover in the passband lost to the feedback, but transformer coupling. When a push-pull power this introduces memory effects into the nonlinear amplifier is operated in Class A biasing, both characteristic. tubes are actively amplifying during the entire signal cycle. Alternatively, Class AB biasing can be used, where one tube handles the signal for Dynamic Operation positive signal excursions while the other tube is in a low current quiescent state, and vice versa for Capacitive elements exist throughout the tube cir- negative excursions. Leaving the quiescent tube cuit, preventing it from being accurately modeled as in a low-power state gives Class AB operation a static waveshaper (a memory-less nonlinearity). If higher power efficiency, but it may also introduce large transients are present in the input signal—as is crossover distortion as the tubes transition between often the case with the —the grid-to- quiescent and amplifying states. Also, because Class cathode voltage could become positive, and current AB amplifiers draw current from the Igk will flow from the grid to the cathode, eventu- proportional to the signal amplitude, large input- ally causing the device to cut off, introducing an voltage bursts can cause a momentary decrease undesirable phenomenon called blocking distortion in the supply voltage. This effect, called sagging, (Aiken 2006). Also, because a grid capacitor is often introduces further compression used to block the direct-current (DC) component (Aiken 1999b).

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2009.33.2.85 by guest on 26 September 2021 The power amplifier is coupled to a guitar response describes how the system reacts to a loudspeaker through an output transformer, which unit impulse. The representation of this introduces additional distortion and hysteresis (i.e., impulse response is known as the an increasing signal is distorted differently than a and describes the gain or attenuation applied to the decreasing signal). Furthermore, the loudspeaker input signal at various frequencies. Once the impulse itself can also contribute significant nonlinear response is known, e.g., on the computer in digital behavior both acoustically and electrically. form, convolution with this impulse response will In conclusion, the complicated interdependen- recreate the effect of this filter. cies and dynamic nonlinearities in vacuum-tube There are two general methodologies of modeling amplifiers make their accurate physical modeling linear systems in guitar circuits. The black-box extremely demanding. As a result, approximate system identification approach views the system models simulating only some of the most noticeable as an abstract linear system and determines coeffi- phenomena have been developed by the amplifier- cients replicating the system. A white-box approach modeling community. derives a discretized frequency response for the system based upon knowledge of its Modeling of Linear Filters in Amplifiers linear, constant-coefficient differential equations. Because the linear systems in guitar amplification are often parametrically controlled (e.g., by poten- To better understand nonlinear distortion modeling tiometers in tone or volume controls), the modeling later in this article, we will first consider the approach must be parametric. simulation of the linear part of the amplifier, namely, the tone stack. The characteristics of linear filtering greatly influence the tonal quality of electric-guitar Black-Box Approach amplifiers. Often, will be provided to allow a to choose between different component In the black-box approach, the linear system is values in a circuit to vary its frequency response. excited with a test signal that covers all frequencies Certain frequency responses are associated with of interest. This signal is usually a frequency sweep particular genres or styles of music and are often of a low-amplitude sinusoidal input or broadband associated with specific guitar-amplifier models. white . A set of measurements is obtained The unique quality of the tone stack of the for various settings of the parameters, which may electric-guitar amplifier is significant enough to be multivariate as for the low, mid, and high tone warrant several attempts in the patent literature knobs of the guitar tone stack. Various techniques to invent methods to make a digital tone-stack are well known for extracting a frequency response model. The tone-stack configurations in guitar from these measurements (Foster 1986; Abel and amplifiers are all very similar. Amplifiers are mainly Berners 2006). differentiated by the component values of the circuit Once the impulse response is found, it can be and the mapping from the controls to these values. used directly as a finite impulse response (FIR) The tone stack typically has up to three knobs filter to simulate the measured system. Because the controlling the gains of three bands, loosely called original systems are typically low-order infinite im- bass, middle, and treble. The middle band is a notch pulse response (IIR) systems, it is computationally in the frequency response. advantageous to identify IIR filters corresponding to the measured response. The digital filter system Digital Filtering identification process optimizes either the error in impulse response (time-domain identification) or A system that introduces no new frequencies frequency response (frequency-domain identifica- to the signal is linear and can be characterized tion) over the set of digital filter coefficients, given completely by its impulse response. The impulse a desired filter order. Preferably, optimizing over the

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2009.33.2.85 by guest on 26 September 2021 impulse response captures phase information and is weights so that the resulting response matches that a simpler, more robust formulation. of the actual circuit. The circuits, which are defined Because the parameterized filter coefficients are using and , are taken into the usually implemented as lookup tables, the patents discrete time domain by the bilinear transform for covering linear modeling of amplifier components digital implementation. generally concern methods to reduce table size and In summary, black-box approaches decide on a storage costs in a practical implementation. The particular filter structure, and then they decide on Fender tone-stack patent (Curtis, Chapman, and coefficients for that structure to match the response Adams 2001) covers an topology that of the target system. Ad hoc mappings from parame- replicates the range of frequency responses of a ter space to coefficient space parameterize the filter. tone stack. Assuming this filter structure, system identification comprises obtaining coefficients for various knob settings by manual tuning to match the White-Box Approach resulting frequency responses. The mapping from pa- rameters to coefficients is compressed for implemen- Yeh and Smith (2006) propose an analytical approach tation by sparse sampling (a suggested five points per to the full tone-stack circuit and suggest that the knob) and 3D linear interpolation of the coefficients. resulting parameter update equations are not pro- The Gustafsson et al. (2004) patent also de- hibitively complicated. This approach derives the scribes multidimensional linear interpolation for full third-order transfer function with no approxi- the compression of mapping from parameters to mations for the filter by symbolic circuit analysis. filter coefficients. This approach improves upon Because the coefficients are described as algebraic the accuracy of classical linear interpolation and functions of the parameters, this method is fully reduces the number of entries needed in the table parametric. Yeh, Abel, and Smith (2007) applied this by warping each parameter dimension using non- approach to filters based upon operational amplifiers. linear mapping functions prior to interpolated table The tone stack for the Boss DS-1 distortion pedal lookup. The patent also describes the decomposition was implemented by interpreting the analog filter as of the resulting filter into a linear combination of a weighted sum of high-pass and low-pass functions Kautz basis filters, a particular form of second-order and implementing the analogous structure digitally. digital filter, for stability in implementation. This is a special case of the general technique in digital Nonlinear Modeling signal processing to ensure numerically stable filter implementations by decomposition into second- Nonlinear signal processing is at the heart of order sections. More information concerning Kautz tube-amplifier modeling. Here, we review static filters in audio applications can be found in Paatero waveshaping with memoryless nonlinearities, and Karjalainen (2003). which is a fundamental technique in digital- A gray-box approach incorporating some insight distortion implementations, and several categories into the structure of the circuit, described in a of methods to reintroduce memory into the patent application by Gallien and Robertson (2007), nonlinearity: ad hoc nonlinear filters based upon divides the tone stack into a parallel bank of two the circuit signal path, analytical approaches, first-order filters, one high-pass and one low-pass, and nonlinear filters derived from solving circuit which are weighted and added. The filters are equations using numerical methods. cleverly devised approximate equivalent circuits comprising resistors and capacitors that allow for implementation of the parameter mapping. The Static Waveshaping equivalent circuits are simulated and compared to a simulation of the full circuit to derive component The most straightforward method for obtaining values for the equivalent circuits and the filter signal distortion with digital devices is to apply an

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2009.33.2.85 by guest on 26 September 2021 Figure 2. Construction of nonlinearities and suitable Figure 3. Solid line: used in Doidic et al. the digital effects device scaling coefficients. The input–output plot of the (1998). Dash-dotted line: described in Araya and amount of distortion can nonlinear function of the asymmetric Suyama (1996). The be varied by changing the Equation 1 used in Araya nonlinearity in Equation 3, distortion block consists of scaling coefficients. and Suyama (1996); dotted also used in Doidic et al. three identical line: the input-output plot (1998). The allowed of the symmetric operation range is denoted nonlinearity in Equation 2, with dashed lines.

Figure 2

instantaneous nonlinear mapping from the input variable to the output variable. This type of alteration is called waveshaping (Arfib 1979; Le Brun 1979). If the mapping does not change in time, this method is called static waveshaping. An early Yamaha patent (Araya and Suyama 1996) describes a digital guitar effects device using this technique. This is illustrated in Figure 2. In Figure 2, the signal from the instrument is first fed to the distortion block through an analog-to-digital (A/D) converter (including an analog amplifier for setting a suitable input level). The distortion effect is obtained by feeding the signal into a nonlinear function through a scaling coefficient. The nonlinear function used in Araya and Suyama (1996) is of the form   3x x2 Figure 3 y = 1 − (1) 2 3 the form where x is the input (bounded between [−1, 1]) and y  is the output signal. The nonlinear curve produced by 3 f (x) =− 1 − [1 − (|x|−0.032847)]12 Equation 1 is illustrated in Figure 3 with a solid line. 4  Because the curve is fairly linear in the operation 1 range of the device, the scaling and nonlinearity is + (|x|−0.032847) + 0.01, 3 applied three times in cascade (i.e., sequentially) for obtaining more distortion. After leaving the distor- for − 1 ≤ x < −0.08905 tion block in Figure 2, the signal is fed to a collection f (x) =−6.153x2 + 3.9375x, of linear effects (e.g., or reverberation) and finally to a digital-to-analog (D/A) converter. Araya for − 0.08905 ≤ x < 0.320018, and and Suyama also suggest adding a digital equalizer f (x) = 0.630035, for 0.320018 ≤ x ≤ 1(3) between the A/D converter and the distortion. More nonlinear functions are suggested in Doidic can be used. Figure 3 illustrates the input-output et al. (1998), including a symmetric function of the curve defined by Equation 2 using a dotted line and form the curve defined by Equation 3 using a dash-dotted f (x) = (|2x|−x2)sign(x)(2)line. It must be noted that the original patent (Doidic et al. 1998) has some typographical errors in the where sign(x) = 1ifx > 0, and sign(x) =−1other- equation of the asymmetric nonlinearity, and thus wise. Alternatively, a hard- function or a it does not produce the input–output relationship piecewise-defined asymmetric static nonlinearity of illustrated in Figure 3.

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2009.33.2.85 by guest on 26 September 2021 As displayed in Figure 3, all the input–output means that instead of applying a nonlinear algebraic curves are fairly linear for small-amplitude signals, function, such as the one in Equation 1, the system that is, signal values near the origin. This obviously reads the input–output relation from a pre-stored means that the smaller the signal is, the less it table, for example, a digitized version of Figure 3. is distorted. A patent by Toyama (1996) uses a The advantage of this technique is that it is easier signal-dependent scaling procedure with a nonlinear to obtain a desired type of input-output relation, function to also distort small-amplitude signals. because the designer can freely draw the input– This technique can add harmonic content to various output curve for the lookup table. signals regardless of their amplitude levels, although On the other hand, a high-resolution lookup it does not resemble the behavior of vacuum-tube table would consume an excessive amount of distortion. A further Yamaha patent (Shibutani memory, so low-resolution lookup tables and 1996) describes a computationally simple method interpolation algorithms must be used. Also, run- for creating piecewise-linear distortion functions by time modification of the nonlinearity becomes branching the signal via various scaling coefficients difficult. Digidesign implemented this type of and adding the output. Graphically, this means that lookup-table waveshaping in their early software each of the scaling coefficients determines a slope Turbosynth in 1989. for a linear segment in the input–output plot. In an early study by Sullivan (1990), a simple non- Another simple digital distortion circuit, “man- linear function or a lookup table is used in distorting tissa fuzz,” is described in Massie (1996). This exotic the output of a synthesized guitar string. In fact, algorithm uses a simple bitshifing operation in dis- the nonlinear function in Equation 1 can be seen torting the input signal. Although the mantissa-fuzz as a scaled version of the one suggested in Sullivan technique is computationally extremely efficient, it (1990). Sullivan’s article also introduces a system for seems virtually impossible to match the distortion simulating the acoustic feedback between synthe- curve to a desired nonlinearity. sized guitar strings, amplifier, and a loudspeaker. Moller,¨ Gromowski, and Zolzer¨ (2002) describe a technique to measure static, nonlinear transfer Oversampling curves from all stages of a guitar amplifier. Their goal is to mimic the nonlinearities and filters in Nonlinear signal processing blocks are known to the signal path of the amplifier, approximating the expand the bandwidth of the incoming signal, nonlinearities as static, the filters as linear, and which in a DSP system can cause aliasing if the neglecting loading between stages. Santagata, Sarti, bandwidth of the output exceeds the Nyquist and Tubaro (2007) introduce a model of the triode frequency (i.e., half the sampling rate). An amplifier preamplifier with an added hard-clipping feature. model can distort harmonic signals such as a This model uses an iterative technique for evaluating guitar tone and produce many new harmonics in the nonlinear tube equations, but it does not the output that, through aliasing into the audio incorporate the capacitive effects of the triode stage; range, are no longer harmonically related to the therefore, it can be considered as computing the original tone. The resulting noisy, “dissonant” implicitly defined waveshaping curve “on the fly,” sound owing to aliasing is characteristic of low- based on parameters measured from an actual tube. cost digital implementations of strong and is typically mitigated through running the distortion algorithm at an oversampled rate, which Lookup-Table Nonlinearity is computationally expensive. Preceding the patent by Araya and Suyama (1996), In the late 1990s, the Company patented a there had already been some studies on how to obtain digital guitar amplifier, i.e., an amplifier and effects digital distortion effects. Kramer (1991) introduced emulator combined with a loudspeaker (Doidic et al. a simple method for obtaining arbitrary nonlinear 1998). This device used a sampling rate of 31.2 kHz distortion in real time using a lookup table. This for most of the signal processing, but it included

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2009.33.2.85 by guest on 26 September 2021 Figure 4. Tube-amplifier at a higher sampling rate modeling scheme, as to avoid aliasing. suggested in the Line 6 Multichannel output can TubeTone patent (Doidic be used, for example, in et al. 1998). The conjunction with stereo nonlinearity is evaluated effects.

an eight-times oversampling circuit for evaluating as , chorus, or . If or line a static nonlinearity at 249.6 kHz, thus attenuating output are used, a simple low-pass filter can also be the aliased distortion components. This straightfor- applied for simulating the effect of the loudspeaker ward technique, named TubeTone Modeling, was cabinet. Finally, the signal drives a loudspeaker (or used in several commercially successful Line 6 several , if for example stereo effects digital guitar-amplifier emulators. are used) after a D/A conversion and amplification. Figure 4 illustrates the system described in Doidic et al. (1998). Here, the digital signal is first fed to Customized Waveshaping a collection of preamplifier effects—that is, effects that are typically located between the guitar and An interesting method for obtaining a highly amplifier, such as a , compressor, or a customized type of distortion has been introduced wah-wah. Next, eight-times oversampling with in Fernandez-Cid´ and Quiros´ (2001). This technique, linear interpolation is applied to the signal, and it illustrated on the left of Figure 6, decomposes the is fed to a nonlinearity. After the nonlinearity, the input signal into frequency bands using a filterbank, signal is lowpass-filtered using an antialiasing FIR and it then applies a different static nonlinearity filter, and it is downsampled back to the sampling for each band separately. Thus, only narrow-band rate of 31.2 kHz. signals are inserted to the nonlinear waveshapers, Figure 5 visualizes what happens to the waveform and the perceptually disturbing and spectrum of a sinusoidal input signal when distortion is minimized. The authors call this tech- distorted by the nonlinear Equations 2 and 3. nique multiband waveshaping. The delay imposed The top row illustrates the waveform (left) and on the direct signal in Figure 6 equals the delay spectrum (right) of a 1.2-kHz sinusoidal signal with caused by the filterbank, so that the signal phase is an amplitude of 0.8. The middle row shows the correctly preserved after the final summation. signal after the symmetric distortion defined by Fernandez-Cid´ and Quiros´ (2001) suggest using Equation 2. As expected, the symmetric distortion Chebychev polynomials as the nonlinearities. These creates a “tail” of odd harmonics in the output polynomials are a special type of function allowing signal spectrum. For frequencies above the Nyquist the designer to individually set the amplitude of limit (a sampling frequency of 44.1 kHz was used each harmonic distortion component, provided here), the harmonics fold back to the audio band, that the input signal is purely sinusoidal with resulting in frequency components that are not in unity amplitude. Furthermore, using this type of any simple harmonic relation with the input tone. polynomial approximation, aliasing can be avoided The bottom row shows the input signal after the for sinusoidal input signals, because the designer heavy-clipping asymmetric distortion defined by can simply choose not to synthesize the highest Equation 3. As can be seen in the lower right graph, harmonics. The right part of Figure 6 illustrates the asymmetric distortion creates even and odd the construction of a single Chebychev-based harmonic components. The upper components are waveshaper used in Fernandez-Cid´ and Quiros,´ again aliased back to the audio band, resulting in an where the overall signal level is set between [–1, 1] inharmonic spectrum. prior to the evaluation of the nonlinearity. Dynamic In Doidic et al. (1998), the output signal from the nonlinearities can be imitated by using two different distortion is fed to a collection of linear effects, such polynomials ( fA(x)and fB(x) in the right part of

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2009.33.2.85 by guest on 26 September 2021 Figure 5. Signal waveforms sinusoidal input signal the input signal after the (left pane) and the with a frequency of 1.2 heavy-clipping asymmetric corresponding frequency kHz; middle row: the input distortion defined by spectra (right pane) for signal after the symmetric Equation 3. sinusoidal input-output distortion defined by signals. Top row: a Equation 2; bottom row:

Figure 6) in parallel, and varying their mix ratio nonlinearities that change their shape according to according to the signal level of the corresponding the input signal or some system-state variables. band. Finally, the original dynamics of the signal An early digital system for emulating a tube are restored by multiplying the polynomial output amplifier was outlined by Pritchard (1991). He with the signal level, as shown in the right part of suggested using two nonlinear distortion blocks Figure 6. The authors claim that the waveshapers with a digital equalization unit in between. Ideally, perform well, even though their input is not a the first distortion block would have a high-pass sinusoid but rather a narrowband signal. filter with the cutoff frequency controlled by the Patents by Jackson (2003) and Amels (2003) input-signal polarity, and an asymmetric static present trigonometric functions for creating static nonlinearity for producing mainly even harmonics. waveshapers where the distortion component levels The second distortion block would generate both can be set by the designer. Schimmel and Misurec even and odd harmonics and emulate the sagging (2007) implemented and analyzed static nonlinear- effect of the power amplifier using a dynamic ities using piecewise-linear approximations of the nonlinearity. Aliasing problems, however, are not nonlinear input-output curves. These three meth- addressed by Pritchard. ods use oversampling to suppress aliasing. Also, A more detailed description of a dynamic tube- a polynomial approximation of a static nonlinear- amplifier model has been discussed in a Yamaha ity without aliasing suppression is presented in patent (Kuroki and Ito 1998). There, a single tube Schimmel (2003). stage is again modeled using a lookup table, but the DC offset of the input is varied according to the input-signal envelope. The authors give the Ad Hoc Nonlinear Filters impression that this bias variation would be caused by grid capacitor charging owing to grid current, Because the assumption that the nonlinearities although a more realistic explanation would be the are memory-less does not hold for describing the variation of the cathode voltage owing to a change in behavior of real tube amplifiers, researchers have plate current. A tube preamplifier can be simulated proposed various dynamic waveshapers, namely, by connecting several tube-stage models in cascade.

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2009.33.2.85 by guest on 26 September 2021 Figure 6. Construction of and the signaling inside an the multiband waveshaper individual waveshaper is distortion, described in depicted in the right half. Fernandez-Cid´ and Quiros´ In the right half, the (2001). The overall output of the averager can structure is illustrated in be seen as a measure of the the left half of the figure, overall signal level.

Sign inversion is applied between tube stages for signal-analysis block estimates the signal energy modeling the phase-inverting behavior of a real tube for the last few milliseconds, and it checks whether stage. Note that, owing to the dynamic nonlineari- the input signal is increasing or decreasing. Next, ties (i.e., signal history-dependent DC offsets), the the polynomial coefficients are interpolated from preamplifier stages cannot be combined as a single a set of pre-stored coefficient values according to equivalent lookup table. A push–pull power ampli- the signal energy. The pre-stored coefficients are fier can be simulated by connecting two tube-stage obtained from measured tube data using system- models in parallel and reversing the sign of the other identification techniques (see, e.g., Nelles 2000). branch. With suitable DC-offset values, crossover The hysteresis effect can be simulated by using distortion can be emulated, if desired. The system a different set of polynomial coefficients for in- proposed in Kuroki and Ito is illustrated in Figure 7. creasing and decreasing input signals. The authors Another dynamic model of a guitar preamplifier suggest implementing the static nonlinearities with has been presented in Karjalainen et al. (2006). This Chebychev polynomials to avoid aliasing, and also model assumes that the plate load of the tube stage is because the accuracy of the Chebychev polynomial constant and resistive, so that the tube nonlinearity approximation is highest near the signal extrema simplifies to a mapping from the grid voltage Vgk (i.e., around ±1, near saturation). to plate voltage Vp. This curve is measured from the tube by shorting the cathode to ground and varying Analytical Methods the grid voltage. Grid current is also measured as a function of the grid voltage. These curves are Several methods exist for analyzing a nonlinearity combined in a single precomputed V -to-V table. gk p with memory. These are based upon Volterra series Bias variation is simulated using a feedback loop, theory and can be used to implement nonlinear as in Kuroki and Ito (1998). The filtering effect audio effects. caused by the grid resistor and Miller capacitance is modeled with a low-pass filter at grid input, Volterra Series while a high-pass filter emulates the interstage DC-blocking filter. Three tube-stage models are The Volterra series expansion (Boyd 1985) is a used in series and connected to a loudspeaker model representation of systems based upon a nonlinear via an equalizer. A minimum-phase FIR filter is expansion of linear systems theory. Analogous to used as a loudspeaker model. convolution with the impulse response vector of a An interesting system-identification-based ap- linear system, the Volterra series is a multidimen- proach has been presented by Gustafsson et al. sional convolution with nonlinear system-response (2004), the founders of the Swedish company Soft- matrices. Whereas in linear systems the impulse re- ube AB (producers of Amp Room software). Here, sponse fully characterizes the system and allows its the dynamic nonlinearity is simulated by feeding output to be predicted given an input, Volterra sys- the signal through a nonlinear polynomial function tems are characterized by special functions, called and varying the polynomial coefficients according kernels, that correspond to the multidimensional to the input signal. Figure 8 illustrates this. The impulse response of the nonlinear terms. It can also

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2009.33.2.85 by guest on 26 September 2021 Figure 7. A dynamic tube with a signal-dependent push–pull power amplifier amplifier model as DC-offset. An entire is simulated by connecting described in a Yamaha preamplifier can be the tube stage models in patent (Kuroki and Ito simulated by connecting parallel and in opposite 1998). The model of a tube-stage models in phase. single tube stage consists cascade with a phase of a lookup table, added inversion in between. A

be regarded as a Taylor series expansion with the Dynamic Convolution polynomial terms replaced by multidimensional Kemp (2006) has patented a black-box method, convolution, accounting for the memory associated dynamic convolution, for nonlinear system analysis with different orders of nonlinearity. and emulation. The basic idea of this technique is Volterra series have been used extensively to simple: several impulses with different model nonlinear acoustic systems including loud- are inserted into the distorting system during the speakers. In particular, they can linearize low-order analysis, and the resulting impulse responses are distortion circuits and loudspeakers in real time (e.g., recorded. System emulation is carried out using con- Katayama and Serikawa 1997). Farina, Bellini, and volution, so that the amplitude of each input sample Armelloni (2001) and Abel and Berners (2006) used is detected and compared to the set of impulse a technique to identify parameters for a subclass amplitudes used in the analysis. Once the near- of Volterra systems based upon a frequency-sweep est measured impulse is found, the corresponding excitation of the system. A similar technique is used impulse response is used in evaluating the convo- in the Nebula effects sampler by Acustica Audio lution. Because this procedure is applied for each (www.acusticaudio.net), which allows the user to input sample, the convolution coefficients change create soft-saturating models of several audio effects according to the input signal level during run-time. based on the system response. Helie` (2006) applied Although a promising technique, dynamic con- a specific Volterra series expansion to create a real- volution has some limitations. First, the amount time effect that includes the third-order nonlinear- of stored data can be prohibitively large if a high- ities of the Moog ladder filter. Schattschneider and amplitude resolution is used. Secondly, dynamic Zolzer¨ (1999) report an efficient implementation of convolution can be used for modeling static nonlin- a type of Volterra series and a system-identification earities, but it fails to model dynamic nonlinearities, technique to derive parameters for their namely, systems for which the shape of the nonlin- model. earity changes due to the input signal (Berners and Although Volterra series are a theoretically Abel 2004). Note that the nonlinear convolution valid black-box method for simulating various introduced by Farina, Bellini, and Armelloni (2001) nonlinearities, real-time emulation of strongly can be seen as the Volterra representation of the saturating distortion poses a problem. This is dynamic convolution method. because Volterra series involve a convolution of a dimension equal to the order of the nonlinearity for each nonlinear term in the model, making the number of coefficients and computational cost Circuit Simulation-Based Techniques grow rapidly with increasing order of nonlinearity. Because guitar distortion often involves very strong, The preceding techniques have all treated the clipping-type nonlinearities, Volterra series are not distortion device as a nonlinear black box, possibly the preferred technology for this application. with memory. Techniques based upon solving the

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2009.33.2.85 by guest on 26 September 2021 Figure 8. A dynamic polynomials are suggested Figure 9. A single stage of Equivalently, this is a amplifier stage model, for implementing the the nonlinear digital Moog nonlinearity with described in Gustafsson et nonlinearities. A complete filter (Huovilainen 2004). embedded memory, al. (2004). The nonlinear amplifier can be simulated The nonlinearity is derived by discretizing the function f(x) is varied each by connecting several embedded within the circuit equations. time sample according to amplifier stage models in digital filter feedback loop. the input signal cascade. characteristics. Chebychev

where N is the number of rows or columns of the square matrix G, it has been found empirically that for typical circuits a sparse LU solve is O(N1.4), owing to the sparse nature of the matrix equations (White and Sangiovanni-Vincentelli 1987). As com- putational power increases and researchers model ordinary differential equations (ODEs) that describe more complex circuits, MNA offers a simple way to the behavior of the circuit have also been attempted. construct circuit schematic-based audio effects.

Transient Modified Nodal Analysis Custom, Simplified Ordinary Differential Equation Solvers design involves the engineer- ing of analog and digital systems based upon For commercial effects, the simplest highly nonlinear integrated circuit devices such as acceptable implementation is desired, because metal-oxide-semiconductor field-effect transistors companies boast of their capability to provide () and bipolar transistors. Verification of a multitude of real-time effects simultaneously. the designs depends critically on the accuracy of To this end, several researchers have developed numerical circuit simulators, e.g., the Simulation effects based on simplifying the ODE model of the Program with Integrated Circuit Emphasis (SPICE; circuit and trading off accuracy for efficiency in the Vladimirescu 1994). SPICE uses transient modified numerical ODE solvers. nodal analysis (MNA) with nonlinear components in Huovilainen reported nonlinear models of the audio circuit simulation. MNA solves the equations Moog ladder filter (2004), as well as operational describing circuit behavior in matrix form, GV = I, amplifier (OTA)-based all-pass where V is a vector containing the node voltages; filters (2005), by deriving a minimal ODE from the I is a vector containing the current contributed by circuit equations and solving it using Forward-Euler the nonlinear devices, capacitors, and sources; and numerical integration. The result is a nonlinear G is the conductance matrix representing the linear recursive filter structure with a nonlinearity embed- current-to-voltage relation of each component in the ded in the filter loop. Huovilainen’s nonlinear Moog circuit. MNA is particularly convenient, because filter model is illustrated in Figure 9. A simplified the computer can easily derive the circuit equations version of this model has been presented in Valim¨ aki¨ given a circuit schematic. and Huovilainen (2006). The matrix G is typically sparse, because it Yeh et al. (2008) extended this approach to encodes the connections between the components strongly clipping diode-based distortion circuits of the circuit, which are typically connected to just and found that for circuits in general, implicit ODE a few neighbors. MNA requires the solution of this methods such as Backward Euler or Trapezoidal equation, usually by LU decomposition. Although Rule are needed to avoid numerical instability at the complexity of a general matrix solve is O(N3), typical sampling rates. Implicit methods require the

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2009.33.2.85 by guest on 26 September 2021 numerical solution of an implicit nonlinear equation Although WDFs are a computationally efficient, by iterative fixed-point methods, a general subclass modular physical-modeling technique—and thus of which are the Newton–Raphson methods. Yeh a promising method for flexible real-time audio and Smith (2008) also extended this approach to circuit simulation—some barriers to widespread the triode preamplifier using a state-space approach application of WDFs remain. Finding a general with a memory-less nonlinearity (the vacuum-tube methodology in the WDF framework to model Ipk expression itself), demonstrating that implicit instantaneous feedback loops between different methods transform the ODEs for audio circuits into parts of the amplifier circuitry presents a significant a recursive state-space structure with a multidimen- challenge. Also, certain circuit topologies, such as sional static nonlinearity embedded in the feedback bridges, do not easily map to connections of the loop. This approach accounts for both the implicit adaptors commonly used for WDFs. nonlinearity of the circuit and the memory intro- duced by bypass, coupling, and Miller capacitances in the circuit. It can be considered a brute-force, Other Models fixed-sampling-rate simulation of the circuit. A recent patent by Gallo (2008), the founder of A hybrid DSP/tube amplifier has been patented by Gallo Engineering (producers of Studio Devil soft- (Suruga, Suzuki, and Matsumoto 2002). Their ware), introduces a tube-stage emulation algorithm system uses an upsampled nonlinear function in using a parametric nonlinear function. The bias modeling the preamplifier, while the power amplifier variation is modeled by evaluating the cathode is emulated using two push-pull triodes, connected voltage ODE using a numerical solver, such as to a solid-state power circuit via a transformer. A the fourth-order Runge–Kutta algorithm. The plate central processing unit (CPU) controls the biasing of voltage variation is neglected here, as in Karjalainen the tubes and the filtering of the feedback from the et al. (2006). output to the input. The power amplifier state can be switched between class A and class AB biasing by the CPU. Furthermore, the solid-state power circuit Wave Digital Filters couples the output transformer to the loudspeaker so Wave digital filters (WDFs; Fettweis 1986) are a that the output power rating can be varied without special class of digital filters with parameters that altering the interaction between the tubes and the directly map to physical quantities. Each of the loudspeaker. Amplification, a subsidiary of basic electrical circuit elements has a simple WDF Korg, manufactures a hybrid DSP/tube amplifier representation, and, through the use of “adaptors,” modeling system called Valvetronix. the resulting filters connect to each other as real A recently introduced exotic sound effect electric components do. Thus, the user can build the (Pekonen 2008) uses a time-varying allpass filter WDF circuit model by connecting elementary blocks in adding phase distortion to the input signal. Al- (resistors, capacitors, etc.) to each other like a real though various types of distortion could be emulated amplifier builder. A real-time model of a WDF tube- by suitably modulating the filter coefficients, the amplifier stage has been presented in Karjalainen current usage of this effect does not allow convincing and Pakarinen (2006). Here, the tube is modeled emulation of vacuum-tube distortion. using a two-dimensional lookup table for simulating the bias variation, while the effect of the grid current is neglected. Sound examples are available Summary and Discussion at www.acoustics.hut.fi/publications/papers/icassp- wdftube. Yeh and Smith (2008) demonstrated that Digital emulation of guitar tube amplifiers is a the WDF can efficiently represent certain guitar vibrant area of research with many existing com- circuits, such as the bright and the two- mercial products. Linear parts of the amplifier, capacitor diode clipper. such as the tone stack, are modeled using digital

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2009.33.2.85 by guest on 26 September 2021 filters, for which the parameters are found with all the fine details and nuances of the vacuum-tube system-identification methods or by using a priori sound, and to make it widely available for use in knowledge of the underlying circuitry. In the sim- artistic expression. plest case, the distortion introduced by the tube stages is modeled using static waveshaping. Aliasing problems can be avoided using oversampling. More- Acknowledgments sophisticated methods can be used for the simulation of dynamic nonlinearities. Most of these methods Jyri Pakarinen’s research is funded by Helsinki Uni- can be classified as being inspired by circuit signal versity of Technology. David Yeh was supported by a paths, which try to model the signal path from the National Science Foundation Graduate Fellowship. amplifier’s input to the output. There are also some The authors wish to thank Prof. Matti Karjalainen, methods that attempt to simulate the operation of Prof. Vesa Valim¨ aki,¨ Miikka Tikander, and Jonte the underlying electric circuit, but these are often Knif for helpful comments. either greatly simplified or still too demanding com- putationally for real-time modeling of complex cir- cuits. Alternatively, some analytical methods, such References as Volterra series or dynamical convolution, have also been suggested. Owing to the complex dynam- Abel, J. S., and D. P. Berners. 2006. “A Technique for ical nonlinearities of the tube-amplifier circuit, true Nonlinear System Measurement.” Proceedings of the Audio Engineering Society 121st Convention.New physics-based models for accurate real-time simula- York: Audio Engineering Society, paper no. 6951. tion of the tube amplifier have yet to be discovered. Aiken, R. 1999a. “What is Miller Capacitance?” Available It must be noted that owing to the essentially online at www.aikenamps.com/MillerCapacitance nonlinear, complex nature of tube amplifiers, objec- .html (accessed Apr. 7, 2008). tive evaluation of their —and hence Aiken, R. 1999b. “What is ‘Sag’?” Available online the sound quality of tube emulators—is extremely at www.aikenamps.com/Sag.html (accessed Apr. 14, difficult. Thus, the best way to rate different emu- 2008). lation schemes is by listening. Marui and Martens Aiken, R. 2006. “What is ‘Blocking’ Distortion?” Available (2002) have presented some studies discussing per- online at www.aikenamps.com/BlockingDistortion ceptual aspects of amplifier modeling. As a result .html (accessed Apr. 7, 2008). of the subjectivity of human listeners, one should Amels, D. 2003. “System and Method for Distorting a Signal.” U.S. Patent No. 6,611,854 B1. Filed Sep. 22, be careful not to underestimate certain amplifier- 2000, issued Aug. 26, 2003. modeling schemes just because the method used is Araya, T., and A. Suyama. 1996. ”Sound Effector Capable simple or physically inaccurate. Careful tuning of of Imparting Plural Sound Effects Like Distortion and the emulation parameters can make a tremendous Other Effects.” U.S. Patent No. 5,570,424. Filed Nov. improvement in the resulting sound. 24, 1993, issued Jun. 4, 1996. Existing emulation techniques are improving in Arfib, D. 1979. “Digital Synthesis of Complex Spectra both physical accuracy and sound quality. Owing by Means of Multiplication of Nonlinear Distorted to the easy distribution of digital media, software Sine Waves.” Journal of the Audio Engineering Society amplifier emulators are also constantly gaining new 27(10):757–768. users. Although some tube-amplifier enthusiasts Barbour, E. 1998. “The Cool Sound of Tubes.” IEEE might feel that digital emulation is a threat to the Spectrum 35(8):24–35. Berners, D. P., and J. S. Abel. 2004. “Ask the Doc- tube-amplifier industry, the authors believe that it tors!” Universal Audio WebZine 2(6). Available on- should rather be viewed as an homage. It can also be line at www.uaudio.com/webzine/2004/july/text/ seen as a form of conservation, because the quantity content2.html (accessed Jun. 10, 2008). and quality of available tube-amplifier components Boyd, S. P. 1985. “Volterra Series: Engineering Fun- continues to dwindle. After all, the ultimate goal damentals.” Doctoral Thesis, University of Cal- of amplifier emulation is to convincingly reproduce ifornia at Berkeley. Available online at www

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